All of the 4th grade teachers and students from Gardner Bullis went on a field trip to an archaeology museum. Tickets were $$7.00$ each for teachers and $$3.50$ each for students, and the group paid $$45.50$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$28.00$ each for teachers and $$11.50$ each for students, and the group paid $$164.50$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+3.5y = 45.5}$ ${28x+11.5y = 164.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-28x-14y = -182}$ ${28x+11.5y = 164.5}$ Add the top and bottom equations together. $ -2.5y = -17.5 $ $ y = \dfrac{-17.5}{-2.5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {7x+3.5y = 45.5}$ to find $x$ ${7x + 3.5}{(7)}{= 45.5}$ $7x+24.5 = 45.5$ $7x = 21$ $x = \dfrac{21}{7}$ ${x = 3}$ You can also plug ${y = 7}$ into $ {28x+11.5y = 164.5}$ and get the same answer for $x$ ${28x + 11.5}{(7)}{= 164.5}$ ${x = 3}$ There were $3$ teachers and $7$ students on the field trips.